We may now generalize the condition of Lemma 2.1 from posets to categories in an obvious way: we define a locally small category C: to be continuous if it has (small) filtered colimits and the functor Ii3 : Ind-4 -+ I( has a left adjoint. (In view of Lemma 1.1, we may thus interpret Lemma 2.1 as saying that a poset is a continuous category iff it is a continuous poset.) Before investigating the consequences of this definition, we give a lemma which will be useful in many cases in verifying the existence of a left adjoint to 15.
Lemma 2.2. Suppose that 8 has filtered colimits and pullbacks and that, for each f : X - Y in 6, the pullback functor f * : J/Y + J/Xpreserves filtered colimits. Then the functor lh_n: Ind-6 4 G is a fibration (in the sense of [ 111).
Proof. Let ( l’&el be an ind-object with colimit Y, and f: X -+ Y a morphism of 6. Writing X; for the pullback Xx, Y, we obtain an ind-object (Xj)iEl, which by hypothesis has colimit X; and the projections X + 8 define a morphism of ind- objects f: (X)iE/+(K),Ef with Ii&n(~) = f. It is easy to verify that $ is a Cartesian morphism (with respect to the functor lin+n), and conversely that an arbitrary Continuous categories and e.rponentiable reposes 265 morphism of ind-objects is Cartesian iff it factors as an isomorphism followed by a morphism of the form f. Hence the Cartesian morphisms of It-id-6 are stable under composition, and so 1% is a fibration. 17
Corollary 2.3. Under the hypotheses of Lemma 2.2, constructing a Ieft adjoint to lim* : Ind-8 -6 is equivalent to constructing an initial object in each of its fibres.
Proof. To construct the adjoint at a particular object X of r’:, we have to find an initial object in the comma category (X11@). But the fact that 1% is a fibration easily implies that an initial object in the fibre over X, together with the identity map from X to its colimit, is initial in this comma category; the converse is obvious. •I
We note that the hypotheses of 2.2 and 2.3 are satisfied either if P satisfies ‘Axiom ABS’ (finite limits commute with filtered colimits) or if A is a topos (in which case the functors f * preserve all colimits). The next result provides us with a plentiful supply of continuous categories.
Proposition 2.4. For any IocaIfy small category 8, the category Ind-A is continuous.
Proof. We already know Ind-6’ has filtered colimits (1.2). The embedding y : t” + Ind-6 induces a full embedding Ind-y : Ind-r! --, Ind-Ind-6; we shall show that Ind-y is left adjoint to li$ : Ind-Ind-6 hind-A. We have already observed that any ind-object (XJie, is the colimit in Ind-A of the objects y(X;), i E I; i.e. the composite I&I .Ind-y is (naturally) isomorphic to the identity. Let (T)iel be any filtered diagram in Ind-B, and suppose T=Qi T is the ind-object (Xj),,~. Then because each y(Xj) is finitely-presentable in Ind-6, the canonical maps Y(Xj)+ T in Ind-6 each factor in an essentially unique way through some 7;:+ T, and so we get a unique map in Ind-Ind-6
(Y(q))j,J + (T)isl whose image under 15 is the identity map on T. It is straightforward to verify that this map is a component of a natural transformation from Ind-ye li,m to the identity functor on Ind-Ind-8, which satisfies the ‘triangular identities’ with the iso- morphism tdtnd_b+ 1% aInd-y. So we have an adjunction Ind-y i Ii@. Cl
Corollary 2.5. Any locally finitely presentable category (in the sense of Gabriel- Ulmer [7]) is continuous.
Proof. It is well known that a locally finitely presentable category d is equivalent to the ind-completion of its full subcategory t”r, of finitely-presentable objects. Cl 266 P. Johnsrow. A. Joyal
For our next result, we need an ‘adjoint-lifting’ lemma which seems not be widely known; so, although it was proved in [21), we repeat the statement of it here.
Lemma 2.6. Suppose given a diagram of categories and functors
in which A ’ is a (pseudo-)retract of (5 (i.e. RI z idT), 3 is a retract of .:‘, and we have isomorphisms TI’zIT’, RTs T’R’ which are compatible with the retraction isomorphisms in the sense that RTI’- T’R’I’ 1I !I RIT’- T commutes. Suppose further that T has a left adjoint L, and that idempotents split in 3’. Then T’ has a left adjoint.
Proof. See [21], Lemma 1.5. 0
In general, the hypothesis that idempotents split in .3’ cannot be omitted; the ‘naive’ construction L’= R’LI yields a functor which is not itself left adjoint to T’, but which has an idempotent endomorphism whose image is the desired left adjoint. However, in the applications which concern us this restriction will not be irksome; for we shall be dealing with categories which possess filtered colimits, and the image of an idempotent may be computed as a colimit over the two-element monoid { 1, e : e2 = e}, which is a filtered category.
Proposition 2.7. Let 6 be a continuous category, and let 8’ be a (pseduo-)retract of E (as in Lemma 2.6) by functors which preserve filtered colimits. Then it is continuous.
Proof. First, the hypotheses imply that 8’ has filtered colimits, since every filtered diagram in 6’ is in the essential image of the retraction R : 8 -+t’. Now we simply apply Lemma 2.6 to the diagram Continuous categories and exponentiable toposes 267
lim Ind-A’- d’
Ind-I
Ind-6’ - ri” which commutes because I and R preserve filtered colimits. G
Theorem 2.8. A locally small category ri’is continuous iff it is a retract of a category of the form Ind-.P by functors preserving filtered colimits.
Proof. If 6 is continuous, the functor 15 and its left adjoint L express it as a retract of Ind-6’ (note that the counit of the adjunction (L il@) is necessarily an iso- morphism, since the unit of (li$ i y) is an isomorphism), and they both preserve colimits since they have right adjoints. The converse follows directly from Proposi- tions 2.4 and 2.7. q
Of course, 2.8 generalizes the characterization of continuous posets as retracts of posets of the form Idl(P) (‘algebraic posets’) by maps preserving joins (‘Scott- continuous maps’). (For continuous lattices, this result is already to be found in [33].) But it is worth noting that the proof of 2.8 actually tells us slightly more than is claimed in the statement; for it shows that an arbitrary continuous category can be embedded as a retract of one of the form Ind-R in such a way that the retraction is right adjoint to the inclusion. That is, if (ignoring problems of size) we write $ for the 2-category of categories of the form Ind-l’:, functors preserving filtered colimits (which we might as well call ‘Scott-continuous functors’) and natural transforma- tions, then the categories which we obtain by splitting (pseudo-)idempotents in R (i.e. the continuous categories) may in fact all be obtained by splitting idempotent comonads. This observation will be of importance when we come to consider injective toposes in the next section. It is natural to ask whether, in a continuous category R, we have some analogue of the way-below relation in continuous posets. Indeed we do; but it turns out that we must regard it not as a ‘relation’ (i.e. a property of certain morphisms of J), but as an additional structure which may be carried by such morphisms. We shall devote the rest of this section to developing it. Let d be a continuous category, and write L : 6 + Ind-A for the left adjoint of le. We define a wavy arrow from X to Y in G (denoted X--, Y) to be a morphism y(X) 4 L(Y) in Ind-A, i.e. an equivalence class of morphisms from X to vertices of the filtered diagram L(Y). We write _%omd(X, Y) (or simply Nom(X, Y)) for the set of all wavy arrows from X to Y. Clearly, we have a canonical map E : .xomb(X, Y)+ Hom,(X, Y) 268 P. Johnsrone. A. Jo_val which sends the equivalence class of a morphism X + Y, (U, some vertex of L(Y)) to the composite X + Y,-+li$; Yia Y. (Thus E is just the functor li$ applied to morphisms y(X) 4 t(Y) in Ind-6.) We shall call e(f) the underlying straighr arrow of the wavy arrow f. However, E is not in general a monomorphism; two different wavy arrows may have the same underlying straight arrow, which is why we must regard ‘waviness’ as a structure rather than a property.
Example 2.9. Let A be the category of (left) G-sets, where G is a group. Then 6 is locally finitely presentable and so continuous by Corollary 2.5; moreover, from the proof of Proposition 2.4 we see that t(X), for a G-set X, is the filtered diagram whose vertices correspond to all morphisms from (a representative set of) finitely- presentable G-sets to X. Now it is easy to see that a G-set X is finitely-presentable iff (a) X has finitely many G-orbits and (b) for each XEX, the stabilizer subgroup G,r= {ge G ) gx=x} is finitely-generated. (We are indebted to R. Borger for this observation.) Suppose G itself is not finitely-generated. Since G is finitely- presentable as a G-set, the two projections G x G- G both define wavy arrows G x G - 1 in A; and these wavy arrows are distinct since the two projections cannot be coequalized by any map from G to a finitely-presentable G-set. But they have the same underlying straight arrow, since there is only one map G x G -+ 1.
Since y and L are functors, it is clear that we can compose a wavy arrow j-:X - Y in 6 with either a straight arrow g : T-+X or a straight arrow h : Y 4 2 (the results being wavy arrows T -Y and X-Z respectively), by forming the composites f’ y(g) and L(h). f in Ind-6. Moreover, since Ind-A is a category it is clear that these two types of composition are associative and commute with each other; and since li,m is a functor the map E converts both types of composition into the ordinary composition of straight arrows. Thus we have proved:
Lemma 2.10. The assignment (X, Y) - .Nom(X, Y) is a profunctor (=distributeur) R-- 38, i.e. a functor 6 Opx 6 + y; and E is a morphism of profunctors from xorn to the unit (Yoneda) profunctor Horn : g--G. 0
A slightly less trivial, but very useful, result is:
Lemma 2.11. Let f : X- Y and g: Y - Z be two wavy arrows in 8. Then the composites g . .z(f) and e(g). f are equal as wavy arrows X-Z.
Proof. Regarding the composites as morphisms y(X)+L(Z) in Ind-6, it is easy to see that each is the composite
f Y(X) - L(Y)2 Y(Y)& L(Z), where i : L( Y) 4 y(Y) is the unique morphism of ind-objects lying over the identity map on Y. 0 Continuous categories and e.uponenriable loposes 269
Lemma 2.11 tells us that we have a well-defined composition for wavy arrows; and it follows easily from the definitions that the eight possible associative laws
(f*g).h=f.(g.h), where each of the arrows f, g, h may be either wavy or straight, are all satisfied. In particular, taking g to be straight and the other two to be wavy, we would be entitled to regard composition of wavy arrows as defining a morphism of profunctors
if only the domain of this morphism were legitimately definable. The trouble is that, since A is in general a large category, .Norn an .r/om(X, Y) is defined as a quotient of the proper class LL ,r/om(Z, Y) x .Fom(X, Z), Zeobd and so we have no right to regard it as a set. However, we observe that every wavy arrow [f] : Z - Y can be factored as
for some vertex Y of I.(Y), and so any composable pair X-Z- Y is equal (as a member of .Nom@,.r/om(X, Y)) to one of the form X- I$- Y, thus in the definition of xom as .Nom(X, Y) we may restrict the variable Z to run over the set of objects I: which occur as vertices of the diagram L(Y). (More explicitly, we may define xorn aa .Fom(X, Y) to be l$i .yom(X, Y).) In this way .fom&.Hom becomes a legitimate profunctor, and we may regard composition of wavy arrows as a morphism of profunctors p as above. Moreover, one of the eight associative laws tells us that ,u is itself associative in an obvious sense.
Proposition 2.12. The morphism of profunctors p : .Nom BA .Nom -.dom defined above is an isomorphism.
Proof. First we show that ,Uis surjective, i.e. that any wavy arrow can be factored as a composite of two wavy arrows. (The argument here generalizes the proof of the well-known ‘interpolation property’ of the way-below relation in a continuous poset - cf. [8], Theorem I 1.18.) Let X be an object of 6; write (X;)iel for the ind- object t(X) and (Xii),E,, for each L(Xi). Since L is a functor, the 15(x;) form a filtered diagram in Ind-6, and by the proof of Theorem 1.2 the colimit of this diagram is a filtered diagram in 6 whose vertices are all the X,j, joU,,, J,. Now the functor lim : Ind-6’ -+I’ preserves colimits, and so Ii$(l@j L(X,)) f IirJl;(l@ J!,(Xj))z ‘$,“I Xi z X* Transposing this isomorphism, we get a morphism of ind-objects L(X) + l&l, L(X,) 270 P. Johnstone, A. Joyal lying over the identity map on X, in particular for each ic I the canonical map li : Xi +X can be factored as
h l;. Xl ___* Xi> -Xiv-X for some i’ and some i. Furthermore, the composite X, + Xi> + Xi, represents the ith component of a map of ind-objects t(X)+L(X) over X, which must be the identity; so this composite represents the same wavy arrow Xi -X as idx,. Thus, given any wavy arrow Y-X represented by f : Y + Xi, say), we may factor it as
[hfl y -.-.-MXi, ------X9Iid1 as required. Next we must show that fl is injective, i.e. that if
is a commutative square of wavy arrows, then the pairs (h, f) and (k, g) are already equal in Nom@, .r/om( W, Z). First, since L(Z) is a filtered diagram, we may represent h and k by morphisms hi : X + Zi and ki : Y -+ Zi for the same index i. NOW the square
need not commute, but. since both ways round represent the same wavy arrow W-Z, we can find i -+i’ in the index category for L(Z) such that Zi+ Zi, coequalizes them. Even so, the square Continuous categories and exponentiable toposes 271 need not commute at the wavy level; but since the two wavy arrows W-Z;. have the same underlying straight arrow, it follows from Lemma 2.11 that they have equal composites with (the underlying straight arrow of) any wavy arrow with domain Z,.. Accordingly, we now use the first part of the proof to factor [idzJ : Z;. - Z as a composite Z;, - Z,. - Z, and replace hi, and kiVby their com- posites with the underlying straight arrow of Z;,-Z;-. We then have a diagram
in which all cells commute at the wavy level, from which we deduce that (h,f) and (Ic,g) are equal in Yom 0, Juom( W, Z). •i
In view of Proposition 2.12, we may consider the inverse of p as a morphism of profunctors .Yom d.xom 0, .Yom. Since p is associative, p(-’ is coassociative; and from the way in which p was defined it is easy to see that E : .#tom + Horn is a counit for ,u-‘. We thus have:
Theorem 2.13. For any continuous category A, the structure (.Nom,p(-‘,e) is an idempotent profunctor comonad. 0
So far, we have not imposed any ‘size restrictions’ on A beyond that of local smallness. However, it frequently happens in practice that a continuous category R, though not itself small, has a small generating subcategory of a particularly nice kind. We next investigate this possibility. Let V be a small full subcategory of 8. We shall say that Y’ is A-filtered if, for every object X of 6, the comma category V/X (whose objects are R-morphisms with domain in ‘6’and codomain X) is filtered. Note that this condition holds if %’has finite colimits which are preserved by the inclusion V -+ 8; but it is not necessary to assume that %’ has finite colimits. We shall write ‘#jX for the category whose objects are all wavy arrows from objects of Y to X, and whose morphisms are commutative triangles of the form
C-D
\/ X
Lemma 2.14. If Y is R-filtered, then V{ X is filtered for every X. 272 P. Johnstone, A. Joyal
Proof. Let L(X)=(Xi)jel. First, VJX is nonempty since the filtered category I is nonempty and V/X, is nonempty for any EE I. Next, suppose we have two wavy arrows f :C---X, g:D- X. Since I is filtered, we can represent f and g by straight arrows into the same Xi, and then use filteredness of V/X; to construct as diagram
C-E-D
\I/ Xi with E in %‘,which we can interpret as a diagram
C-E-D
‘\,I/ X
The verification of the third condition for filteredness is similar. cli
We recall that a full subcategory %’of A is said to be dense ([25], p. 241) if every object of R can be expressed as a colimit of objects of ‘6’.Of course, if such an expression exists, there is a canonical one: we can express X as the colimit of the forgetful functor Ux : %/X -+A which sends (f : C-X) to C.
Lemma 2.15. If %’is dense in A, then any object X of Y is expressible as the colimit of the forgetful functor ‘l/x : “5 X -A.
Proof. Consider a cone I under the diagram #x (with vertex Y, say). For each vertex Xi of the diagram L(X), we may construct a cone Ai under Uxi; specifically, if f: C-X;, we define (Ai), to be At/), where [f] is the wavy arrow C-X represented by f. So by density of %’we obtain unique factorizations vi: Xi+ Y of each of these cones through the colimiting ones. From the uniqueness, it is clear that the vi themselves form a cone under t(X), and so define a unique map v : li$ t(X) IX + Y. So the canonical cone under tix with vertex X is a colimiting cone. 0
Suppose now that ‘6 is both dense and (y-filtered. Then for any object X of 6, we can regard +I~as an object of Ind-A’ with colimit X. Moreover, from the definition of wavy arrows it is clear that there is a morphism of ind-objects from 11~to f.(X), whose fth component (for f : C--X an object of %‘jX) is f itself, and which lies over the identity morphism on X. But by the universal property of L(X), we must Conrinuous caregories and exponentiable roposes 273
also have a unique morphism of ind-objects L(X)+ ;“x over the identity on X, and the composite f.(X)- i’lx-L(X) must be the identity. Consider the composite tix+L(X)- ‘l/x. If f: C-X is any object of zSX, then by the first half of the proof of Proposition 2.12 we may factor f as C -!A D LX (where there is clearly no loss of generality in supposing that D is an object of %); and then c(h) : f +g is a morphism of ‘6jX, so that if k : D -, tix(l : E --+X) represents the gth component of the above composite, then k- e(h) represents its fth component. But the diagram
must commute since this morphism lies over the identity on X; hence
k c(h) C *E
s I ‘\I X commutes at the wavy level, i.e. km I is a morphism of %jX. But this means that the given endomorphism of tix is the identity, and so tix is isomorphic to L(X) in Ind-4. Furthermore, it is not hard to see that this isomorphism is natural in X, if we make X - ‘Nxinto a functor A + Ind-A in the obvious way; and so we have proved:
Proposition 2.16. Let A be a continuous category. Then the left adjoint L : I’ - Ind-(5 of li,m may be taken to factor through Ind- %4 Ind-A, where % is any small, full, dense, J-filtered subcategory of A. Cl
In view of 2.16, we obtain a refinement of Theorem 2.8:
Corollary 2.17. The following conditions on a category 6 are equivalent: (i) d is a retract, by filtered-colimit-preserving functors, of a category of the form Ind-%’ where V is small. (ii) t” is continuous and has a small, fuil, dense, J-filtered subcategory.
Proof. (ii) * (i): If %’is such a subcategory, then the functors
L lim 8 - Ind-%’ and Ind-V- Ind-A A/$ express d as a retract of Ind-%; and they preserve filtered colimits since the inclusion Ind-V + Ind-R is the ind-extension (in the sense of Lemma 1.3) of % - 4. (i)= (ii): It is easy to see that %’is dense and Ind-Y-filtered in Ind-z; and its image under a retraction Ind-V-6 has the same properties relative to A. E 274 P. Johnstone. A. Jo_val
As we noted in the case of Theorem 2.8, the first half of the proof above actually tells us rather more than is claimed in the statement: namely that any A satisfying (ii) is embeddable as a coreflective subcategory of some Ind-‘6. But for a small category %, there is no harm in identifying Ind-q with its image in the functor category [ %“P, .Y ] (since every object of this category has a canonical representation as a small colimit of representables); as usual, we shall call a functor ‘6”P*.Y flat if it is (isomorphic to) a filtered colimit of representables, and write Flat( x04 .Y) for the full subcategory of flat functors. (It is well known [4] that if % has finite colimits, then the flat functors ‘6‘P*.Y are just the finite-limit-preserving ones.) If % is a subcategory of a continuous category A as in 2.17(ii), it is naturally of interest to have a characterization of the coreflective subcategory of Flat( ‘6‘P, Y) which is the image of R under this identification. Of course, the flat functor X”P--r.r/ which corresponds to the ind-object ‘l/x is just (the restriction to % of) the functor .Yom(-, X).
Proposition 2.18. With ‘6’and (5”as in Corollary 2.17, a flat functor F: ‘6°P+.Y is isomorphic to one of the form .Wom(-, X) (X an object of (‘) iff it satisfies the folio wing condition: (*) For every object C of %’ and every XEF(C), there exists a wavy arrow f : C-D (with D an object of W) and y E F(D) such that x = F(ef )( y).
Proof. If F is the functor .Yom(-,X), then condition (*) follows from the ‘sub- divisibility’ of wavy arrows, i.e. the first half of the proof of Proposition 2.12. (As we have already remarked, there is no loss of generality in requiring the object in the middle of the factorization to lie in the subcategory Y.) Conversely, suppose (*) is satisfied; then (identifying F with an ind-object in W) we wish to show that the counit map p: L(l$ F)+F is an isomorphism. Let (Qiel be the ind-object corresponding to F (note that the indices i are just the elements of LL cEob(F(C)), and let X be its colimit in R. The map /I is defined as follows: given a wavy arrow f : C--X, choose a representative h : C + C, for the fth component of the unique morphism of ind-objects ti~-+(CJiEI over X, and then define J&-(f) = F(h)(i) (where we regard the index i as an element of F(Q). It is straightforward to verify that this is well defined, and a natural transformation of functors. It is easy to see that p is surjective; for if XE F(C), then by (*) we can find f: C-D and y E F(D) mapping onto x, and then the composite
is an element of .Yom(C,X) which is mapped by /I to x. (Here ,Iv denotes the yth component of the colimiting cone.) But in fact the above construction yields a well- defined natural transformation rz : F + .Yom( -, X), which is a one-sided inverse for p; to see this, it is sufficient to prove that it is well defined, since naturality is then Continuous categories and e.uponenriable toposes 27s obvious. Suppose we have f: C-D, g: C -E, y E F(D) and z E F(E) such that F(af)(y) =x= F&)(z); then by flatness of F we can find morphisms h : D --*B, k : E + B in ‘6 and c E F(B) such that
commutes. In particular, the’ composites h -f and k. g: C-B have the same underlying straight arrow; but by the argument already given to prove surjectivity of b,n, underlies some wavy arrow B --X, and so by Lemma 2.11 the composites I, - he f = AYef and 13,. k. g = 1;. g are equal as wavy arrows. Thus o(x) is a well- defined wavy arrow C-X. So we have expressed the functor F as a retract of .Nom(-, X) in the category of flat functors ‘6’OP+ Y’ whose corresponding ind-object has colimit X in 6; but since ~Vom(-, X) is initial in this category, it has no proper retracts, and so is isomorphic to F. 0
Proposition 2.18 tells us that a continuous category 6 can be reconstructed (up to equivalence) from the pair (‘6’.T), where %’is a generating subcategory of I? as in 2.17, and T: %‘--+V is the profunctor obtained by restricting Zotn : c”--36. As we have already remarked, the proof of idempotency of .~orn which we gave in Proposition 2.12 remains valid if we restrict the objects involved to lie in ‘6; so Tis still an idempotent profunctor comonad. Moreover, T is leftfiuf in the terminology of [24], i.e. the functors T(-,C): ’C OP + 3’ are all flat. (Equivalently, the functor (-) 0, T: [ %, Y] + [V, Y’] preserves finite limits.) In the converse direction, note that a left flat profunctor T: ??--+I/ between small categories is essentially the same thing as a functor I/ -F1at(%‘OP, U)z Ind-‘6; and hence essentially the same as a filtered-colimit-preserving functor Ind-9 -+ Ind-%: So the bicategory Re of small categories, left flat profunctors and morphisms of profunctors is equivalent (contravariantly at the level of l-arrows) to a full subcategory of the 2-category $ considered after Theorem 2.8, namely that whose objects have the form Ind-‘6’ where V is small. Hence if we split the idempotents (or more particularly, the idempotent comonads) in Be, we obtain a full subcategory of the idempotent-completion of P, namely the 2-category of continuous categories satisfying the size restriction of 2.17. The passage from (‘6’.T) to the subcategory of flat functors satisfying (*) is clearly functorial, and extends the passage from V to Ind-V, so it is the required embedding of the idempotent-completion of Re in that of R. A curious side-effect of Proposition 2.18 is to tell us that a left flat, idempotent 276 P. Johnsfone. A. Joyal
profunctor comonad T on a small category % is determined up to isomorphism by its image under the counit map E : T + Horn u; for in order to state the condition (*) we need only know which straight arrows of ‘6 underlie wavy arrows, and not what the wavy arrows themselves are. It is not at all clear ab initio why this should be so.
3. Injective toposes revisited
In an earlier paper [21], the first author investigated a notion of injectivity for Grothendieck toposes (more generally, for bounded .%toposes, where .Y’ is an arbitrary base topos). In that study, the structure of idempotent profunctor comonad played an important role, for reasons which were not entirely clear. The appearance of the same structure in our investigation of continuous categories is the key which enables us to open up the link between the two concepts, generalizing the link which Scott [33] discovered between injective spaces and continuous lattices, and incidentally clarifying the status of the two conditions which appeared in [21] as unwarranted assumptions. We shall continue to assume for notational purposes that our base category .Y is ‘the’ topos of constant sets, but in practice it could easily be generalized to any topos with a natural number object, by rewriting our arguments (which are all constructive) in the language of categories indexed over .I/’ [31]. First we recall one of the main results of [21]:
Proposition 3.1. A bounded Stopos is injective (with respect to sheaf subtopos inclusions) iff it is a retract in 23Zop/.Y of a functor category [P’, Y] where ‘6 has finite limits. 0
It will be convenient for the time being to broaden our considerations to include all retracts in 23~op/.V of presheaf toposes; we shall call them quasi-injective. (It is not clear whether there is in fact any injectivity condition which characterizes these toposes; it is interesting to note that Hoffmann [14] has characterized the corresponding class of spaces by a projectivity condition.)
Proposition 3.2. Let 3- be a quasi-injective topos. Then 2- has enough points, and its category of points is continuous and satisfies the size restriction of Corollary 2.17.
Proof. Since .Y is a retract of some [V “9 Y], it is in particular a surjective image of [VP, 9’1; but presheaf toposes always have enough points. Now the category of points of a Grothendieck topos has filtered (sindexed) colimits by [20], Corollary 7.14; and from the proof of that fact, it is easily deduced that these colimits are preserved by the functors 232op/9’(.5/:8) -*92op/2(.~ 3) induced by geometric morphisms 8 -, 5 So the category of points of 9 is a retract, by filtered-colimit- Continuous cafegories and exponenriable roposes 277
preserving functors, of 23?op/.?(.Y, [d’ “9 ?‘I); but the latter is equivalent by Diaconescu’s theorem [4] to Flat(V, .Y) - i.e. to Ind-% OP. 0
We note in passing that the Lowenheim-Skolem theorem for points of Grothendieck toposes ([20], Theorem 7.16) ensures that if 8 is the category of points of an arbitrary Grothendieck topos, then it has a small, full, dense, E-filtered subcategory. In the converse direction, let t’ be a continuous category satisfying the hypotheses of 2.17. We wish to construct a quasi-injective topos 9 whose category of points is equivalent to A. Although we shall see eventually that .Y may be constructed directly from 8, in order to establish its basic properties we shall need to work in terms of a particular generating subcategory ‘5’of A as in 2.17. Let T, as before, denote the restriction to %’of the profunctor 3~0m on r(‘. Since T is left flat, we can regard (-) 0, T as the inverse image of a geometric morphism t : [‘6’, .‘/I -+ [ ‘6, .i/] over .‘/ (which is of course idempotent, since T is). We shall also wish to refer to a particular Grothendieck topology Jr on PP determined by T, as follows: a cosieve on an object C of %’is Jr-covering iff it contains the underlying straight arrows of all wavy arrows with domain C. The fact that Jr is a Grothendieck topology follows easily from the known properties of T; in particular, the ‘local character’ axiom (T2) of [12], II 1.l is implied by the idem- potency of T.
Proposition 3.3. Let %’ be a small category, and T a left flat, idempotent pro- functor comonad on %. Then the image (in the topos-theoretic sense) of the geometric morphism t : [ ‘5, .‘/I + [Y, .Y] induced by T is Shv( %‘P,JT), where JT is the Grothendieck topology defined above. Moreover, Shv(‘/: ‘P, JT) is also the image oft in the idempotent-splitting sense; in particular, it is a quasi-injective topos.
Proof. To identify the image of t, we have to determine which subobjects R + Horn&C, -) (i.e. which cosieves on C) are mapped to isomorphisms by the functor (*=(-)a, T. But
R@,T*Hom,(C, -)a, T= T(C, -)
is an isomorphism iff, for each wavy arrow f : C-D, there exists g : C -+ E in R and h:E- D such that h - g=f. Clearly this condition implies that e(f) is in R for every such f; but conversely if R contains all the e(f), then we may factor f as a composite C -!. E AD and then c(g) E R. So the covering sieves on C are precisely the Jr-covering ones. Now let
[Y, .Y] 2 Shv( V”P,JT.) 1 [ ‘3 .‘/I denote the (topos-theoretic) image factorization of t; to show that it is also a splitting of the idempotent t, we must show that ri is isomorphic to the identity map 278 P. Johnsrone, A. Joyal on Shv(V’P,Jr). or equivalently that if F is a Jr-sheaf then Fzt,(F). But f,(F) = Tr?l, F, where Th,(-) denotes the right adjoint of (-)@,T, from which we readily deduce the formula
r,(F)(C) = l$,:cW, F(D), the inverse limit being taken over the category whose objects are all wavy arrows with domain C, and whose morphisms are commutative triangles of the form c
i\ D-E
Now the assertion that F is a Jr-sheaf tells us that the canonical map
is an isomorphism, where R is the minimal Jr-covering cosieve on C, i.e. the category whose objects are all underlying straight arrows of wavy arrows with domain C. It is easy to see that this inverse limit maps monomorphically into the one above, i.e. that the canonical natural transformation F + t,(F) is mono. To show it is an isomorphism, we need to show that if x=(x~)~:~,_~ is any element of li$ : cwD F(D), and f,g are two wavy arrows with the same underlying straight arrow, then we must have xJ=xg. But f and g are coequalized by any wavy arrow with domain D, and so xf and xg must have the same image in t,(F)(D)= li$+ F(E). Hence by what we have already proved, xJ=xg; i.e. x is in the image of the canonical map F(C)+f,(F)(C). Cl
Remark 3.4. In the case when %’has finite colimits, Proposition 3.3 was proved in [21] under the additional hypothesis that the ‘underlying straight arrow’ map E was a monomorphism. In view of Example 2.9 and the proof above, it now appears that this additional assumption was unjustified; however, without it we cannot charac- terize the topologies Jr on V”P which arise from profunctors T as in 3.3, as simply as we did in [21], Lemma 2.3. (Conditions (i) and (ii) of the characterization given there remain valid, but (iii) holds only for products and not for arbitrary pullbacks, and there does not seem to be any simple way of reconstructing T from Jr.)
Proposition 3.5. Let 6 be a continuous category satisfying the size restriction of Corollary 2.17. Then there exists a quasi-injective topos 3 whose category of points is equivalent to A. If in addition R has finite co/imits (and is thus cocomplete), then .Y- may be taken to be injective.
Proof. Let %’be a small, full, dense, t-filtered subcategory of R, and let T denote the restriction to ‘6 of the profunctor .r/‘om on 6’. Define .F to be Shv( ‘6“4 Jr), where JT is constructed from T as in 3.3. Then 3 is quasi-injective by 3.3, and by [20], Continuous categories and exponentiable toposes 279
Proposition 7.13, its points correspond to flat functors %“‘P+ju which are ‘continuous’ for the topology JT, i.e. send JT-covering sieves to epimorphic families. But since every object of g has a smallest JT-covering cosieve (viz. the set of all underlying straight arrows of wavy arrows with the given object as domain), it is sufficient to check the continuity condition for these minimal sieves - and for them, it is precisely the condition (*) of Proposition 2.18. So the equivalence d P 232op/Y(x Y) follows directly from 2.18. In the special case when (I”has finite colimits, we may choose our generating sub- category V to be closed under finite colimits in d (in which case it is certainly &-filtered, as we remarked earlier); then [F, Y] is injective by [21], Proposition 1.2, and hence so is its retract .F. 0
To complete the circle, it remains to show that a quasi-injective topos is deter- mined up to equivalence by its category of points. But we already know this fact for presheaf toposes; for we have !ZZJ2op/.Y(.Y;[ ‘6’, 5’1) s Ind-%‘, and by 1.5 we can recover V (or at least its idempotent-completion, which is sufficient to determine the functor category [V, 9’1) from Ind-V as the full subcategory of finitely-presentable objects. And this result extends easily to retracts:
Theorem 3.6. The functor 3~ 232op/.Y(.Y; .X) is an equivalence of 2-categories between the full subcategory of 232op/.V consisting of quasi-injective toposes, and the 2-category &ont of continuous categories satisfying the hypotheses of 2.17 and Scott-continuous (i.e. filtered-colimit-preserving) functors between them.
Proof. If we restrict to presheaf toposes and to continuous categories of the form Ind-x’, then we have an equivalence (the inverse functor being described above). And it is straightforward to verify that any equivalence between 2-categories extends (essentially uniquely) to an equivalence between their idempotent-completions. 0
Corollary 3.7. Any quasi-injective topos is expressible as the image of an idem- potent comonad on a presheaf topos. In particular, any injective topos is expressible as the image of an idempotent comonad on a presheaf topos [ ‘e”P, Y] where V has finite limits.
Proof. We know that the corresponding assertion holds in Qont, by the remarks after Theorem 2.8 and Corollary 2.17; so we may transfer it across the equiva- lence of Theorem 3.6. The particular case follows from the general one as in the proof of 3.5, since the category of points of an injective topos is cocomplete ([21], Corollary 1.7). 0
Corollary 3.7 tells us that the first of the two unsupported assumptions which were made in Section 2 of [21], that we could restrict our atttention to idempotent profunctor comonads, was in fact justified, although (as we have seen) the second was not. 280 P. Johnstone. A. Joyal
There remains one question of interest: we have seen that a continuous category d (satisfying the conditions of 2.17) determines a quasi-injective topos 3 up to equivalence, but the only method we know for constructing 3 involves making an arbitrary choice of a generating subcategory of G. Can we construct 3 directly from 6, without making any such choices? The answer is yes; but is not immediately apparent from the nature of the construction that it always yields a topos, which is why we chose to give a more roundabout, but more explicit, construction first.
Proposition 3.8. Let 6 be a continuous category satisfying the hypotheses of 2.17. Then the full subcategory Q&(6, 9’) of the functor category [6, U] whose objects are Scott-continuous functors is a quasi-injective topos. and its category of points is equivalent to C.
Proof. First we note that Qont(R, 9’) is closed in [6, .Y] under finite limits and arbitrary (small) colimits; so any Scott-continuous functor f: 6 -6 induces a functor f * : &onf(8’, 9’) * &ont(d, 9) which preserves finite limits and all colimits. In particular, if the domain and codomain off * are Grothendieck toposes, then it is the inverse image of a geometric morphism. But in the case when 6 = Ind-V, it is clear that a functor defined on A is Scott-continuous iff it is isomorphic to the ind- extension of its restriction to ‘6, and so Qont(6, .Y)= [ ‘6, .Y’] is a (quasi-injective) Grothendieck topos whose category of points is equivalent to 6. The result for a general d now follows from the fact that any functor preserves images of idem- potents. 0
4. Exponentiable toposes
As indicated in the Introduction, our main objective in this paper is to charac- terize the exponentiable objects in the 2-category %2op/Y of bounded .Xtoposes (where we shall continue to assume for notational purposes that Y is the topos of constant sets). Given toposes 6 and 3 (bounded over .Y), we shall say that the exponential 3’ exists if the category-valued functor 23Xop/.‘/((-) x:, 6, .Y) is representable (in the up-to-equivalence sense), the representing object being denoted X8. We say I(’is exponentiabfe if J” exists for all Y; the operation (-)” is then a (pseudo-)functor %2op/Y + 82op/~, right pseudo-adjoint to (-) X, 6. Similarly if we keep .X fixed and allow A to vary, we obtain a contravariant functor .?(-) defined on the full subcategory of exponentiable toposes in 232op/=r/. We shall make frequent use of the well-known equivalence between bounded .Y-toposes and first-order geometric theories in the laguange of .Y (see [27]). As an example, we begin with a simple but useful lemma: Conrinuous categories and e.rponenriable roposes 281
Lemma 4.1. For any small category ‘6, the functor category [z, .7 ] is exponentiable in 932op/.1.
Proof. For any .Y-topos ~0 the pullback (5 x, [‘;, .‘/ ] is equivalent to [ /, 61 by Diaconescu’s theorem [4]. But by Wraith’s characterization of lax colimits in 20~ [36], [8, “1 is the tensor of r: with % in 20~; that is, we have %2op/.‘/([x, r;], .~)=(%;%2op/.‘r(CC, Y)] for any ./’ Accordingly, if .F is the classifying topos for a geometric theory U, we may define .Ft’g ’ 1to be the classifying topos for the theory [ ‘6,T] whose models are diagrams of type K in the category of U-models. (It is straightforward to construct a presentation for this theory from one for T; see [20], Example 6.55(v).) Z
Lemma 4.2. Exponentiability is a local property; that is, (i) if 6 is exponentiable then so is d/Xfor any X, and (ii) if b/X is exponentiable and X has global support in A, then 6 is e.uponenti- able.
Proof. (i) By a special case of Lemma 4.1, we know that 8/X is exponentiable in %20pM. But a standard argument on exponentials (cf. [20], Exercise 1J) shows that (f: X--6) is exponentiable in 82op/~: iff the pullback functor along f has a right adjoint
rl, : Q2op/Y -+ 232op/Cr. Since the latter condition is clearly stable under composition of bounded geometric morphisms, it follows at once that if 6 is exponentiable in 932op/./ then so is (Y/X. (ii) If X has global support in 6, then the diagram
is a universal coequalizer diagram in 232op/.r/’ (this follows from the descent theorem for open surjections [25], but can in fact be proved much more simply). So the exponential S8, if it exists, should be the equalizer of the diagram
But exponentiability of e/X implies exponentiability of A/XxX, by the first part; hence we can simply define 5” to be this equalizer. 0
Lemma 4.3. A retract (in %2op/Y) of exponentiable topos is exponentiable.
Proof. Suppose G is a retract of an exponentiable topos A’. The idempotent endo- morphism of 8’ whose image is 6 induces, for any ,Y, an idempotent endomorphism of Pr’. If we split this idempotent (which we may do since 82op/.Y has finite limits), we obtain the exponential x6. 0 2a2 P. Johnsrone. A. Jo.vul
In [17], M. Hyland showed that a locale X is exponentiable (in the category of locales) iff the exponential Sx exists, where S is the Sierpinski locale. As we indicated in the Introduction, the underlying reason for this is that S is the free object on one generator in the opposite of the category of locales [18]; in 9335op/.~/, the corresponding role is played by the object classifier .I/ [X] [24]. We now embark on the proof that existence of ?[X]” implies exponentiability of 6; for technical reasons it will be convenient to divide it into two stages.
Lemma 4.4. Let 4 be a topos for which the exponential .r/[X]# exists. Then for any small category ‘6; the exponential .Y [Cl” exists, where .‘/ [C] is the classifying topos for the theory of diagrams of type ‘6.
Proof. By Lemma 4.1, we can regard .r/ [C] as the exponential .I/[X]tKut; and general exponential nonsense shows that (.Y[X] tcYl)d, if it exists, should be equiva- lent to (9 [X]“)t% ut. But the latter topos exists by another application of 4.1. Cl
Theorem 4.5. A topos 6’ is exponenriable in BSop/.r/’ iff the exponential .‘/ [Xl” exists.
Proof. let .i be an arbitrary bounded .P-topos, and suppose it classifies a geometric theory T. Using the techniques of [35], we may present T in such a way that its models appear as diagrams of a certain type % satisfying certain axioms, each of which says that a particular morphism constructed ‘geometrically’ from the diagram is an isomorphism. (For example if T is the theory of objects with a single binary operation /?, we may present it as the theory of diagrams of type
A& wB subject to the axiom that (zt, rr2) : A *B x B is an isomorphism.) Now each such geometric construction, when applied to the generic diagram of type ‘e, gives rise to a geometric morphism where .Y[2] is the morphism classifier over 9: and the statement that the construction applied to U-models yields an isomorphism means that the composite factors (up to isomorphism) through the ‘diagonal’ inclusion .Y[X] --Y/2] which classifies the identity map on the generic object. More particularly, we have a puil- back diagram in d2op/.Y of the form Continuous categoriesand exponentiable reposes 283
,- - II,.7 [X] ‘i I .Ir [C] - r&.r/ [2] where A is an index set for the axioms in the given presentation of U. But the toposes .Y[Cl, I&.‘/ [X] and f&9 [2] all classify theories of diagrams of a certain type (with no additional axioms); so by Lemma 4.4 we can exponentiate each of them to the power rs’provided Y[X]” exists. Also, the functor ( -)G, being a right adjoint, should preserve pullbacks; so we may now define .F” to be the pullback of (&.r/ [Xl)”
.Y [a=]d - (f7,2 121)” and verify that it has the required universal property. •1
So far in this section we have not invoked the theory of continuous categories or of injective toposes. But, as we indicated in the Introduction, there is a very simple link between these topics and exponentiability:
Lemma 4.6. If 6 is exponentiable in 232op/.Y; then it is a continuous category.
Proof. From the definition of the theory it classifies, it is clear that the object classifier 9[X] is injective in the sense considered in Section 3. Also, the functor (-) x,8 preserves inclusions (cf. [20], Proposition 4.47), so its right adjoint (-)” preserves injectives; thus YIXIQ, if it exists, is an injective topos. But we have
6 I232OP/;1(6, .Y’[X])=932op/Y(X Y [Xl”); thus 8 is equivalent to the category of points of an injective topos. The result as stated follows from Proposition 3.2. 0
In the converse direction, suppose 6‘ is a Grothendieck topos which is a continuous category. Then 6’ has a small dense subcategory, which we may take to be closed under finite colimits and therefore &-filtered; so it satisfies the size restric- tion of Corollary 2.17, and by Proposition 3.5 it is thus equivalent to the category of points of a (uniquely determined) injective topos .E Clearly, 3 is the only possible candidate for the exponential 9’[Xlb; to prove that it is the exponential, we have to extend the known equivalence 232op/.Y(.Y,X) = A into a natural equivalence 82op/?(y”, 3) = .Y x, c” for all bounded 5toposes (y : Y-Y). 284 P. Johnsrone. A. Jo.val
Before embarking on this, let us fix some notation. ‘/;’will denote a small (full) generating subcategory of A, which we shall assume closed under finite colimits and limits in 6 (so that in particular ‘6 is a pretopos ([20], Definition 7.38) with arbitrary coequalizers). On ‘6 we have the Grothendieck topology K induced by the inclusion ‘6-t ,I (i.e. the collection of all sieves in % which are universally effective- epimorphic in 6 ), and also the left flat profunctor T obtained by restricting .Fom on A (from which we obtain a topology Jr on 7 Oras in 3.3). Now we have two different ways of embedding G into Ind-%‘=Flat(z “9 .I/); the first sends an object X to the K-sheaf Hom(-,X), and the second sends X to the Jr-continuous functor Xom(-,X). The two embeddings are respectively right and left adjoint to the functor Q : Ind-‘6’46. (There is an interesting parallel between the existence of these two alternative representations of 6 and the confusion in the early days of sheaf theory ([9], pp. 4-5) about whether sheaves should be defined in terms of sections over open sets or closed sets.) To establish the desired natural equivalence, we have to show that the equivalence between K-sheaves and Jr-continuous flat functors remains valid when we replace the category % and the topologies K and JT by their ‘pullbacks’ along a (bounded) geometric morphism y : .Y+Y. Now the pullback ()I*%‘,TK) of a site (‘6’.K) along a geometric morphism y was described in detail in [22], in the particular case when %’ is a semilattice; the general case does not involve any additional complications other than notational ones. In particular, although it is in general not possible to describe YK explicitly in terms of K, we note that if K is the topology generated by a given (pullback-stable) family of sieves on z’, then YK is generated by the image of this family under y*. Thus it follows at once from the definition of Jr that we have P(JT) ‘J(r’T,. Next, we need to describe how the topology K may be generated from the profunctor T. First, since ‘6’is closed under finite colimits in G, we know that K contains the precanonical (finite-cover) topology P on V. Given this information, it now suffices to say which fiffered sieves on objects C of g (i.e. filtered full sub- categories of U/C) are K-covering, since an arbitrary sieve R on C is covering iff the filtered sieve
{f:D+C13afinitecover {gi}iofDs.t.eachf.giER) is covering. But from the definition of wavy arrows, every filtered K-covering sieve on C contains the sieve
Mc={f :D+C]Z?g:D--Cwitheg=f} which is itself K-covering since c” is a continuous category. (Note that Mc itself is not necessarily filtered, if E fails to be a monomorphism, but this does not matter for our purposes.) We thus conclude:
Lemma 4.1. The topology K on q is generated by the precanonical topology P and the sieves MC, C E ob V; in particular a presheaf on % is a K-sheaf iff it is a P-sheaf and satisfies the sheaf axiom for the sieves MC. Continuous categories and exponenriable roposes 285
Proof. The first statement follows immediately from the remarks above. For the second, we have to show in addition that a presheaf which satisfies the sheaf axiom for the Mc also satisfies the axiom for their pullbacks along morphisms of ‘6’.But this is obvious, since if f : D +C is such a morphism then we have MD C f *MC. 0
Now the finite cover topology on a pretopos is preserved by inverse image functors (cf. [22], Lemma 4.2); so it follows from Lemma 4.7 that jjK may be generated from y*T in the same way that K is generated from T. And the assertions “‘6 is a pretopos with arbitrary coequalizers” and “T is a left flat, idempotent pro- functor comonad on ‘6” are both expressible in geometric language, and so preserved by inverse image functors. So we are reduced to proving the following statement: “Given a pretopos ‘6 with arbitrary coequalizers and a left flat, idempotent profunctor comonad T on ‘6, let K and Jr be the topologies on % and %‘P constructed from T as in 4.7 and 3.3 respectively. Then the category of K-sheaves on ‘6 is equivalent to the category of flat, Jz-continuous functors
yoP,,y*”/ Unfortunately, this statement does not appear to be true in general, because the hypotheses are not sufficient to ensure that K-sheaves are flat functors on ‘6OP. The embedding of ‘6 into Shv(x,P) (and hence the canonical functor % -t Shv(Y, K)) preserves coequalizers of equivalence relations - this follows from the fact that each regular epimorphism in ‘6 generates a P-covering sieve - but in order to ensure that arbitrary coequalizers are preserved, we need the additional information that each equivalence relation in ‘6 is covered by the finite powers of any (reflexive, symmetric) relation which generates it. Since such a cover is necessarily filtered, giving this information about our topology K is equivalent to giving the following information about T:
(*) Suppose S is a reflexive, symmetric relation on an object C of %, and let R = U,,, , S” be its equivalence closure. Then every wavy arrow D - R factors through some finite power S” of S.
Thus the pair (X; T) satisfies the condition (*) iff the canonical functor % -+Shv(X, K) preserves all coequalizers, iff Shv( 6, K) is contained in (and therefore a reflective subcategory of) Flat( %“P, .‘/ ). In particular if (z, 7) is derived from a continuous topos A as originally envisaged, then condition (*) is satisfied.
Lemma 4.8. Let Y be an internalpretopos with arbitrary coequalizers in a topos .Y; and T a profunctor on % satisfying (k). Then for any geometric morphism y: Y’d.7, the pair (y*%, y*T) satisfies (*).
Proof. The statement of (*) is expressible (internally) in geometric language, except for the reference to equivalence closures. But since inverse image functors preserve 286 P. Johnstone, A. Joyal natural number objects, they also preserve equivalence closures whenever these exist (as they do in a pretopos with coequalizers). 0
We are now ready for the key step in the proof of our main theorem.
Lemma 4.9. Let V be a pretopos with arbitrary coequalizers, and let T be a left flat, idempotent profunctor comonad on % satisfying (*). Let K and Jr be the topologies on %’and Y”P constructed from T as in 4.7 and 3.3 respectively. Then Shv(%‘,K) is equivalent to the category of points of Shv( V”P,JT).
Proof. As indicated above, the assumption (*) ensures that Shv(Y, K) is a reflective subcategory of Flat(z “4 9). The category of points of Shv(% “4 JT) may also be regarded as a full subcategory of Flat( %‘“P,Y ), namely the category of Jr-continuous functors. Since T is left flat, the functor TO,(-) : [V”P, Y/‘] + [‘COP,9’1 maps Flat( Y”P,9’) into itself; and it inherits an idempotent comonad structure from T. By Proposition 3.3, we know that the image of T on Flat(‘~OP,.Y) is exactly the sub- category of Jr-continuous functors, since it corresponds to the image of the geometric endomorphism of [‘L’,9’1 induced by (-) 0, T. As a functor on [V”P,.Y’], T&(-) h as a right adjoint Tftl c (-), which may be defined by
Clearly, Th,(-) has an idempotent monad structure, and its image consists pre- cisely of those presheaves on %’which satisfy the sheaf axiom for the sieves h4c defined before Lemma 4.7. But it follows from left flatness of T that Thy(-) preserves sheaves for the precanonical topology; so its image on Flat (X “4 9) is just the category of K-sheaves. Now it is easy to see that the adjunction between TQ,(-) and T fk 4 (-) (as functors from Flat( ‘;OP,f 9) to itself) is itself idempotent, and hence that it restricts to an equivalence between Shv(V, K) and the category of Jr-continuous functors. 0
At last we are ready to put together all the ingredients.
Theorem 4.10. A bounded Stopos G is exponentiable in 92op/%’ iff it is a continuous category.
Proof. One direction is Lemma 4.6. Conversely, if A is a continuous category, it suffices by Theorem 4.5 to construct the exponential .Y[X]‘. We define this exponential to be Shv(VOP,JT), where %’is a generating subcategory of 8, closed under finite limits and colimits, and T is the restriction to V of the profunctor Nom on rs’.Now let (y : Y-Y) be an arbitrary bounded .%topos. Working in the context of .Y’-indexed categories, we have equivalences Continuous categories and exponentiable toposes 287
9’ x,8 = Shv( y*V, jjK) =2320p/Y’(.Y,Shv(y*%0P, pJr)) by Lemma 4.9 = 23~op/.Y’(.V’,Y x,Shv( V”P,Jr)) from which we deduce
820~/.7(3” x, 6.9 [Xl) = Z.Qep/yYy”, Shv( ‘#Or’,Jr)). So Shv( W’, Jr) has the required universal property. q
5. Local compactness and exponentiabiiity
In view of the main theorem of the last section, it is clearly of interest to find conditions on a site of definition for a topos A which are equivalent to A being a continuous category. In this section we tackle the problem in the particular case when rF’is localic (i.e. generated by subobjects of 1); it seems likely that some of our methods will extend to more general sites (using the techniques of [5]), but we shall not pursue the matter here. Somewhat surprisingly, in view of the close formal similarity between our main theorem and Hyland’s characterization of exponentiable locales, it turns out that not every exponentiable (=locally compact) locale generates an exponentiable topos of sheaves. However, there is a clear implication in the other direction:
Lemma 5.1. Let A be an exponentiable topos. Then for any object X of 8, the lattice of subobjects of X is continuous.
Proof. If A is exponentiable, then so is A/X by Lemma 4.2; in particular the exponential [2, .Y]“x exists, where [2, Y] is the Sierpinski topos over 5” [20,4.37(iii)]. But [2, U] is a classifying topos for subobjects of 1 in YLtoposes; so it is easily seen to be injective, and hence [2, Y’]“x in injective by the argument used in proving Lemma 4.6. But the category of points of this topos is equivalent to the poset of subobjects of X in 8; so the latter is a continuous pose& and hence (since it is a lattice in any case) a continuous lattice. Cl
Specializing to the case when G is the topos of sheaves on a sober space X, and taking the object X in the lemma to be the terminal object of &, we deduce that if 1$ is exponentiable then the open-set lattice of X must be continuous - equivalently [l, 151, X must be locally compact. However, exponentiability of Shv(X) implies more than local compactness of X. To explain why, we need to introduce a strengthening of the way-below relation: we shall write LIa V (for U, V open subsets of X) if, for every filtered diagram (Fi)isI of sheaves on X with 15; F;z V, there exists icl such that F; has a section over I/. If Shv(X) is a continuous category, this is equivalent to saying that there is a wavy 288 P. Johnsfone. A. Joyal arrow U- C’ in Shv(X), since L(V) (if it exists) is initial in the category of ind- objects with colimit V. The definition of e just given is unsatisfactory in that it refers to arbitrary (filtered diagrams of) sheaves on X, and not just to the open-set lattice of X. Later on, we shall give a characterization of 4 entirely in terms of open sets; for the present, we note merely that it implies the relation 4 (take the F; to be a directed family of subobjects of 1), and that in an important special case this implication can be reversed. We recall that a locale is said to be stably locally compact [23] if it is a continuous lattice and, in addition, the way-below relation is stable under binary meets - i.e. ff, -4 V, and Uz+ Vz together imply CJ,fl &a VI n Vz. Examples of stably locally compact locales include all coherent locales and their retracts, and all locally compact regular locales.
Propostion 5.2. In a stably locally compact locale, UG V implies Ua V.
Proof. Let (FJial be a filtered diagram of sheaves with colimit V. If aFi denotes the support of F;, i.e. the image of the unique map F; -+ 1, then the oFi form a directed family of open se!s with join V, and so we can find icl such that UC OFi - in fact, using the subdivisibility of Q, we can even achieve U< OFi. Now GFi is covered by the open sets over which Fi admits a section, and hence by the sets {WI @IV’s W)(F admits a section over IV’)}, so we can find a finite subfamily { IV,, . . . , W,,} of these sets which covers U. For each Jo n, let si be a section of F;:over an open set Wj’B Wj. For j# k, the two maps
need not be equal, but they become equal when composed with the canonical map Fi + I$ F; 3 V. SO if Ei,+ Wj’fI WL denotes the equalizer of their composites with a map Fi + Ff of the filtered diagram, then the E,, form a directed family of subsets of Win Wi whose join is the whole of Wife Wi. But by stability we have WjfI Wk4 H$‘fl Wi, so there exists i’ with qfl W,C Ei, - i.e. such that the two composites Conrinuous categories and exponentiable roposes 289 are equal. Repeating this argument for each pair (j,k), we arrive at an Fi” and a family of sections W, -F;” which are pairwise compatible - so that they can be patched together to obtain a section of F,- over UT=, W;. But the W; cover U, so we have a section of F;- over I/. q
Example 5.3. If the way-below relation is not stable, then the conclusion of Proposition 5.2 is false. Let X be the space obtained by identifying two disjoint copies of the unit interval [0, l] along the open subspace (0,l) (cf. [34], Example 73); then it is easily seen that X is compact and locally compact, but not Hausdorff. For Ott < 1, we may similarly define X, to be the space obtained by identifying two copies of [0, l] along [0, t); then for f < t’ there is an obvious local homeomorphism X,-X,,, and thus we can regard the X, as a directed diagram of sheaves over Xi =X. Moreover the colimit Ii+, X, is homeomorphic to X; but none of the X, admits a section over X. So we have XaX (by compactness) but not XeX. It may be shown that if L/i and U, denote the two copies of [0, l] embedded in the space X of Example 5.3, then we do have 1/i GZX and CJ2~ X. But U, U U, = X; thus the relation e, unlike Q, is not in general stable under finite joins.
We shall say that a locale X is metustably locally compact if every open V C X can be covered by open sets U satisfying Ua V. Thus Proposition 5.2 tells us that stably locally compact locales are metastably locally compact.
Lemma 5.4. Let X be a locale such that Shv(X) is a continuous category. Then X is metastably locally compact.
Proof. Given V, let L(V) = (FJiEr. As in the proof of Proposition 5.2, we know that V is the join of the supports OFi, iel; and each OF, can be covered by open sets U over which Fi admits a section. But if F; admits a section over U then I/+ V. 0
Our main objective in this section is to show that the converse of Lemma 5.4 is true. Before embarking on this, however, we give our promised example of a locally compact space X such that Shv(X) is not exponentiable. It should be thought of as an ‘iterated’ version of the space of Example 5.3.
Example 5.5. Let X be the quotient of the space [O,l] x 2N (where 2” denotes the Cantor space) by the equivalence relation R, where
(x,s)R(y,r)ox=y, and either s = t or there exists n such that x< 1- l/2”+’ and sln=tl,, (here s(,, denotes “the first n terms of the sequence s”). In terms of its canonical projection onto [0,11, X has a discrete 2”-point space as its fibre over each point of the interval [l - l/2”, 1 - l/2”+‘), and a copy of Cantor space as its fibre over 1. It is 290 P. Johnslone. A. Jqval not hard to see that the quotient map 4: [0, l] x2” -+X is an open map, and hence that X is locally compact. However, we shall show that no open set U satisfying UaX meets the fibre over 1. For if U is a neighbourhood of the point q(l,s), then there exists n such that U also contains the points q( 1 - l/2” + ‘,s) and q( 1 - l/2” + ‘,s’) where s’ differs from s at the (n + 1)st term but not before. Thus U contains an open subspace U’ which looks like the effect of identifying two copies of an open interval (I-E, t + E) (where t=1-1/2”+‘and0<~c1/2”‘*)a1ongthesubinterva1(f-&,f).For0<~’<~,1etX,~ be the space obtained from X by ‘unglueing’ these two intervals over [t-e’, I). Then it is clear that the X,., e’>O, form a directed system of sheaves on X with colimit X. But none of the X,, admits a section over U’, let alone over U, so we do not have uax. Thus we have shown that X is not metastably locally compact; hence by Lemma 5.4 Shv(X) is not a continuous category, and so by Lemma 4.6 it is not exponentiable.
We now embark on proving the converse of Lemma 5.4. First we note that metastable local compactness is a local property (i.e. it holds for X iff it holds for each member of a covering family of open subspaces of X), and thus if X is metastably locally compact, so is the domain of any local homeomorphism E-+X. Thus we may reduce the problem of constructing the functor L : Shv(X)+ Ind-Shv(X) (i.e. of constructing L(E) for every such E) to that of constructing the particular ind-object L(X); and for this it suffices by Corollary 2.3 to construct an initial object in the category of ind-objects with colimit X. We define a category Y as follows: its objects are diagrams (F-t G) where G is a sheaf on X such that for every filtered diagram of sheaves (Hi)ie, with 1%; H~z X, there exists a morphism G *Hi for some i, and F is a subsheaf of G satisfying Fe G in the (continuous) lattice of subsheaves of G. A morphism f: (F, + GI)-+(F2-+ Cl) in % is a sheaf morphism f : F, 4 F2 for which there exists some g : G, --*G2 making the diagram f F, - F 2
1 1 g G,-G 2 commute. As it stands, the category K is obviously not small; but we can cut down to an essentially small full subcategory x0 by imposing the additional restriction on objects that G can be covered by a finite number of sections, i.e. there exists an epimorphism
where the r/l are subobjects of 1 in Shv(X). Conrinrtous curegories and exponenfiable reposes 291
Lemma 5.6. The category ‘LOis filtered.
Proof. It is clearly nonempty. Given two objects (Ft-G,) and (Fz-+G& we may form the coproduct (Ft + F2-+ G, + Gz). Then in the lattice of subobjects of Gt + Gz we have F,~G,IG,+G~ and F2~GzsGI+G2, whence F,+F24GI+Gz; and similarly if we have a filtered diagram (H,)iEj with Ii@; HiaX, then morphisms G, *H, and Gz+ H,, may be combined to produce G, + Gz+ H,. for some i”EI. Also, G, + Gz may be covered by the disjoint union of the finite sets of sections which cover G, and Gz; and the coproduct inclusions F, + F, + F2 clearly define morphisms (F, -, GJ +(F, + F2+ G, + G2) in %,,. Now suppose we are given a parallel pair of morphisms f,, fi: (F, +G,)-+ (Fz+ G2) in ‘4,. Form the diagram fi f3 6 :F2-F 3
III g3 I G I-2-G-G 3 gl where the top row is a coequalizer and the right-hand square is a pushout. By a well- known property of pushouts in a topos [20, Corollary 1.281, the morphism F3~G3 is mono, and the square is also a pullback. In addition, g3 is epi, so the pullback functor gf : Shv(X)/Gs* Shv(X)/G2 is faithful. Now if we are given a directed family (e)rel of subsheaves of G3 with join Gs, then we have vE,gf(U,)=G2, and so there exists iel with g;F3zF2sgj*U,. But then faithfulness of gf tells us that F~s U;; so we have shown that F3 fi h FI- -F-G-Hi-H 2 2 I’ f2 are equal. But then we can factor F2 *Hip through the coequalizer f3, i.e. we obtain a commutative diagram
‘h ’ Gz- Hi -Hi* 292 P. Johnsrone, A. Joyal and hence a factorization Gs -H;, through the pushout. So (Fs + Gs) is an object of ‘6 - in fact of ‘doIsince GJ is a quotient of G2 - and f3 : (Fl- G2) -, (F3;3-G3) is a morphism of r. coequalizing f, and fi. IJ
We have an obvious forgetful functor T: co +Shv(X) sending (F-G) to F, in view of Lemma 5.6, we may regard this as an object of Ind-Shv(X).
Lemma 5.7. If X is metastably locally compact, then rhe colimit of the ind-object T defined above is isomorphic to X.
Proof. X may be covered by open sets CJfor which there exists a V with I!J< VLemma 5.0. Under the hypotheses of Lemma 5.7, the ind-object T is initial among ind-objects with colimit X.
Proof. Let (HJisl be an arbitrary ind-object with colimit X, and let (F-G) be an object of ‘80. By definition, there exists a map h : G -t Hi for some ie I; there may be many such maps, but if hl and h2 are two such, then there exists a morphism Hi -, H, of the filtered diagram such that the equalizer of hl GB H;-H, h2 contains F. Hence the restrictions to F of h, and h2 are equivalent as maps into the filtered diagram; that is,there is a unique equivalence class of maps F-H; which includes the restriction to F of some morphism defined on G. Moreover, it is clear that if we take this distinguished equivalence class for every object (F + G) of %‘c, we obtain a morphism of ind-objects T * (Hi)ie I. We must show that this is the unique such morphism. But if (F-G) is any object of Vo, we can find a subobject F’ of G with FeF’eG, and then we have a morphism (F-+G)+(F’+G) in Vc. So the equivalence class of maps Fd H; assigned to the object (F + G) by a morphism of ind-objects T+(Hi)i,, must contain some morphism which extends to F’; and by the argument given above, this is sufficient to determine it uniquely. 0
Putting together the results of the last four lemmas, we have proved: Conrinuous categories and exponenriable toposes 293
Theorem 5.9. Let X be a locale. The topos Shv(X) is a continuous category iff X is metastably locally compact.
Proof. One direction is Lemma 5.4. In the converse direction, Lemma 5.8 tells us that there is an initial ind-object with colimit X. and Corollary 2.3 says that this ind-object must be L(X); and as explained after Example 5.5, the stability of the hypotheses under localization then ensures that L(F) exists for every sheaf F on X. 0
Corollary 5.10. If X is a stably locally compact space (for example a locally compact Hausdorff space), then the topos Shv(X) is exponentiable in 2332ov/.?. cl
It remains to give a more locale-theoretic (i.e. less sheaf-theoretic) description of the relation a. Now in Chapter 2 of [5], K.R. Edwards investigated the condition that the global section functor Shv(X)-.r/’ preserves filtered colimits (which in our terminology is simply the assertion that X&X); by adapting her characterization to our more general context, we obtain
Proposition 5.11. Let U and V be open sets in a totally compact locale X. Then We V iff the following condition holds: (+) Given any open cover ( Vn)crEAof V, there exists a finite B c A and open sets U, C V, n U (a E B), WaDC U@fl UP (a, /3 E B) such that Wall” V, n VPfor each (a, p) and the canonical diagram
c w,,= & UC-l-* U lz.PEE is a coequalizer in Shv(X). [Informally, U may be constructed by patching together the members of a finite refinement of ( QurE over sets which are way below the pairwise intersections of the V,.]
Proof. First we show the necessity of (+). Given an open cover (Va)aE.4 of V, consider the set of all sheaves F which can be formed as coequalizers
where B runs over all finite subsets of A and Tap4 V,n Vs for each (a, p). We can make these into the vertices of a directed diagram, in which there exists a morphism F-F’ iff (in the obvious notation) we have B c B’ and TODc T$ for each (a, /I) E B x B; and since each V,n Vb is covered by open sets which are way below it, it is easily verified that the colimit of this diagram is V. So if I/e V, there exists a morphism h : U-F for some such F, then we merely define U,= h*( V,) and Wolp=h*(T,s) to obtain the desired properties, since coproducts and coequalizers are preserved under pullback in Shv(X). The proof of sufficiency of (a) is similar to the argument used in proving 294 P. Johnstone, A. Joyal
Proposition 5.2. Let (H,),,, be an arbitrary filtered diagram of sheaves with colimit V, then we can cover V by open sets V, over each of which some If; admits a section s,. Choose open sets U, and W,, as in (+));since we now have a finite set B of indices to deal with. We may assume that the sections S, (a~ B) al! take values in the same Hi. By an argument we have used several times before, we may now find for each pair (a, p) a morphism Hi -+ H? of the filtered diagram such that the equalizer of
contains Wap; choosing H;+Hi. so that this happens for all pairs (a, /I). simultaneously, we deduce that the composite
factors through the coequalizer CaEB U,,- U of C Wab Z C U,. So His admits a section over U, and hence II4 V. E
As stated, the condition (+) of Proposition 5.11 does still make reference to sheaves on X. The trouble is that we cannot in general require the diagram
to be a kernel-pair as well as a coequalizer; that is, we cannot demand that WuD= U,fl U, for all (a, p). It is possible to make C WoD reflexive and symmetric as a relation on C U,; but when we try to take its transitive closure, we face the problem that (in the absence of stability) the relations Wape V,fl VPand Way< Vbfl V, do not imply Wapfl Wpye V,fl Vy. If we wish to remove all mention of sheaves from the condition (+), we can do so by making explicit what it mans for the transitive closure of 1 Wup to be the kernel-pair of C U, + (/; i.e. we may replace the last clause of (a) by the condition: For each pair (a: p), U,,n Up is covered by the sets
~.p,nw,,.,,n-..nW,“_,.,nnW,,.B where (VI, YZ,. . . , y,) (n ~0) runs over all finite strings of members of the index set B.
Acknowledgements
The research described in this paper was carried out at various times and places Conrinuous cafegories and e.rponentiable roposes 295
between May 1980 and April 1981. Both authors are grateful to the Department of Pure Mathematics and St. John’s College, Cambridge, for providing the financial support which made possible the second author’s visit to Cambridge in April-May 1980. We are also indebted to Martin Hyland, whose paper [17] on exponentiability in locales was the spur which prompted us to think about the problem; and to Kathy Edwards, whose painstaking work [5] on the preservation of filtered colimits helped us to avoid some of the less obvious pitfalls in Section 5.
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